Archimedean semigroup

Let $S$ be a commutative semigroup. We say an element $x$ divides an element $y$ , written $x \mid y$ , if there is an element $z$ such that $xz = y$ .

An Archimedean semigroup $S$ is a commutative semigroup with the property that for all $x, y \in S$ there is a natural number $n$ such that $x \mid y^n$ .

This is related to the Archimedean property of positive real numbers $\mathbb{R}^+$ : if $x, y > 0$ then there is a natural number $n$ such that $x < ny$ . Except that the notation is additive rather than multiplicative, this is the same as saying that $(\mathbb{R}^+, +)$ is an Archimedean semigroup.